Place warm water, yeast and sugar in a mixer bowl and stir to combine. Let sit for 5 minutes or until foamy. (You must have the foamy consistency or your yeast isn't working right) See picture above.
Add melted butter, oil, salt and egg and mix briefly in mixer(with mixing attachment). Add flour and continue to mix until the dough begins to form.
Scrape around the sides and attach the dough hook. Knead dough for about 3 minutes until smooth.
Transfer dough to a lightly floured surface and knead until you can form a smooth ball. Divide dough according to the size buns that you desire. I divided it into eight balls which made average size hamburger buns. (see pictures above)
Preheat oven to 425 degrees F.
Cover your baking sheet with a piece of parchment paper. Shape each ball into a bun by pinching the sides with your fingers and pulling the dough up and towards the center, finishing by pushing it into the middle of the bun. Continue all the way around the bun to for a nice round yet slightly flat piece of dough. (see pictures) Place bun onto baking sheet with the smooth side up about 3 inches apart.
Cover the tray with a clean kitchen towel and let rest in a warm place. (I use my warming drawer on "Proof" or turn the oven on to 200 degrees and then turn it off before placing the buns in to rest) Allow to rest 10-15 minutes. (If for some reason they aren't quite as big as you would like after this rest, you can always let them rest longer, checking every 10 minutes for size)
Remove towel, brush with water/egg white mixture and bake in 425 degree oven for 8-12 minutes or until golden brown.
Cool on a wire rack. Brush with melted butter if desired.
